The birational-geometry tag has no usage guidance.

**34**

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### Surfaces in $\mathbb{P}^3$ with isolated singularities

It is classically known that every smooth, complex, projective surface $S$ is birational to a surface $S' \subset \mathbb{P}^3$ having only ordinary singularities, i.e. a curve $C$ of double points, ...

**20**

votes

**1**answer

1k views

### Rationality of intersection of quadrics

Let $X \subset \mathbb{P}^n$ be a complete intersection of two quadrics. It is classical that, if $X$ contains a line, then it is rational. The proof is very simple and basically it is given by taking ...

**19**

votes

**3**answers

2k views

### Birational invariants and fundamental groups

In pondering this MO question and people's efforts to answer it, and recalling also something that I learned in my youth about using Morse theory ideas to prove some results of Lefschetz in the ...

**16**

votes

**3**answers

755 views

### What is known about the birational involutions of P^3?

Describing the group of birational automorphisms of $\mathbb{P}^n$, $\mbox{Bir}(\mathbb{P}^n)$, for $n\ge 3$ is a fundamental open problem in birational geometry. For $n=2$, the classical theorem of ...

**12**

votes

**2**answers

977 views

### is the Hodge conjecture birationally invariant?

Let $X$ and $Y$ be two birational smooth projective varieties over the complex numbers. Assume $X$ satisfies the Hodge conjecture.
Is it known that the Hodge conjecture holds for $Y$?

**12**

votes

**0**answers

749 views

### Relative canonical divisors

Suppose that $X$ is a Gorenstein variety and that $\pi : Y \to X$ is a birational map of varieties with normal $Y$.
In this case the relative canonical divisor is defined to be $K_Y - \pi^*K_X$ (if ...

**11**

votes

**2**answers

516 views

### Geometrically unirational varieties that are not unirational

By a variety over a field $k$, I mean a scheme that is separated and
of finite type over $k$. I indicate changes of the base ring by
subscripts.
Does there exist a smooth and projective variety $V$ ...

**10**

votes

**6**answers

643 views

### When does $Aut(X)=Bir(X)$ hold?

Let $X$ be a projective complex manifold. Under what condition do we have the equality $Aut(X)=Bir(X)$? Here $Aut(X)$ denotes the group of holomorphic automorphisms of $X$ and $Bir(X)$ the group of ...

**10**

votes

**3**answers

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### Contracting divisors to a point

This is quite possibly a stupid question, but it is pretty far from what I normally do, so I wouldn't even know where to look it up.
If $X$ is a projective variety over an algebraically closed field ...

**10**

votes

**2**answers

652 views

### Rationality of GIT quotients

I recently worked through most of the proof of the rationality of the moduli of genus 3 curves, which seemed to have the following structure:
Every nonhyperelliptic genus 3 curve is a smooth plane ...

**10**

votes

**2**answers

388 views

### Blow-ups of $\mathbb{P}^{n-3}$ and $(\mathbb{P}^1)^{n-3}$

Let us consider the points
$$p_1=[1:0:...:0],p_2 = [0:1:...:0],...,p_{n-2} =[0:...:0:1],\\
p_{n-1}=[1:1:...:1]\in\mathbb{P}^{n-3}$$
and the blow-up $X = Bl_{p_1,...,p_{n-1}}\mathbb{P}^{n-3}$.
...

**9**

votes

**3**answers

1k views

### Extending birational isomorphisms between planar curves to the P^2

Let $k$ be a field and let $C,D$ be two integral curves in $\mathbb{P}^2_k$. Now let $f:C \to D$ be a birational isomorphism. Can $f$ be extended to $\mathbb{P}^2_k$. To be precise, does there exist a ...

**9**

votes

**2**answers

849 views

### Blowups of Cohen-Macaulay varieties

Suppose that $X$ is a Cohen-Macaulay normal scheme/variety and $\pi : Y \to X$ is a proper birational map with $Y$ normal.
Question: Is $Y$ also Cohen-Macaulay? Are there common conditions which ...

**9**

votes

**2**answers

303 views

### Fibrations of projective varieties

Let $f:X\rightarrow Y$ be a flat morphism of normal projective varieties with fibers of positive dimension (in particular all the fibers are connected and of the same dimension).
Let $g:X\rightarrow ...

**9**

votes

**2**answers

499 views

### Minimal model which is necessarily singular

I was told during a summer school on the MMP a nice example (which I have also mentioned here on MO) that I'm not able to figure out anymore.
The example (due, I think, to Miles Reid) is a smooth ...

**9**

votes

**1**answer

582 views

### Why is the standard flop a flop?

I have seen at least two ways to define flops (and similarly flips).
We start with $Y \to X$, a surjective birational morphism, contracting a locus of codimension at least 2, such that $K_Y$ is ...

**9**

votes

**1**answer

192 views

### Generalization of the rigidity lemma in birational geometry

Let $X,Y,Z$ be projective varieties, and let $f:X\rightarrow Y$, $g:X\rightarrow Z$ be dominant morphisms. Assume that all the fibers of $g$ have the same dimension and are connected.
If there exists ...

**9**

votes

**1**answer

289 views

### Rationality of moduli spaces of rational curves

Let $\overline{M}_{0,n}$ be the moduli space of Deligne-Mumford stable pointed rational curves, and let us consider the quotient $\widetilde{M}_{0,n} = \overline{M}_{0,n}/S_n$.
Clearly, there is a ...

**9**

votes

**1**answer

466 views

### Schemes as a model category

I'm just learning some basics of model categories, so please forgive me if my question turns out to be trivial. I hope it does at least make sense.
A natural temptation is to relate this machinery to ...

**9**

votes

**1**answer

317 views

### Conditions for the contractibility of subvarieties

One often finds statements of the sort "and one can contract this subvariety $E\subset X$ to a point in the projective variety $W$," without any explanation of the reasons such a contraction exists, ...

**9**

votes

**1**answer

751 views

### Why is proving fully-faithfulness of an integral functor locally analytically sufficient?

More than once I've come across a statement in a paper about derived categories in which it says something to the effect of "in order to prove that $\Phi:D^b(X)\rightarrow D^b(Y)$ is fully-faithful we ...

**8**

votes

**4**answers

2k views

### Is the Euler characteristic a birational invariant

Suppose that $X$ and $Y$ are smooth projective varieties which are birationally equivalent. I would like to have that $$\textrm{deg} \ \textrm{td}(X) = \textrm{deg} \ \textrm{td}(Y).$$ Invoking the ...

**8**

votes

**3**answers

469 views

### Birational automorphisms of canonical models

Let $X$ be a variety with canonical singularities such that $K_X$ is ample.
Do you have a reference of the fact that every birational map from $X$ to itself is biregular?
Thank you

**7**

votes

**1**answer

543 views

### Are groups in (Var/k, rational maps) necessarily algebraic groups?

Let $RVar$ be the category which has complex algebraic verieties as objects and rational maps as morphisms [Edit: for $RVar$ to be a category, the rational maps have to be dominant].
Let's consider a ...

**7**

votes

**2**answers

554 views

### Why is $\mathbb{Q}$-factoriality not local in the étale topology?

I was reading Kollár and Mori's book today and stumbled on the following passage:
"The $\mathbb{Q}$-factoriality assumption is a very natural one if we start with smooth varieties, and it makes many ...

**7**

votes

**1**answer

789 views

### Appropriate journal to publish a determinantal inequality

I have recently made the following observation:
Let $v_i := (v_{i1}, v_{i2})$, $1 \leq i \leq k$, be non-zero positive elements of $\mathbb{Q}^2$ such that no two of them are proportional. Let ...

**7**

votes

**2**answers

808 views

### How much can small modifications change the nef cone?

First let me give a precise formulation of the question; I'll give some background/motivation at the end.
If X is a projective variety which is Q-factorial (meaning X is normal, and some sufficiently ...

**7**

votes

**1**answer

1k views

### A characterization of the blow-up

It is well-known that there is a universal characterization of blow-ups. However, this one is not so easy to use. According to Hartshorne-Theorem II.8.24. The blow-up of a nonsingular $X$ along a ...

**7**

votes

**1**answer

414 views

### Cones, monoids, and the space of (very) ample divisors

An interesting and useful tool to study a projective variety is its ample cone. Understanding the structure of this cone reveals information about the variety, and it is an isomorphism-invariant so ...

**7**

votes

**1**answer

439 views

### Simplified treatment of resolutions of complex analytic varieties?

According to the article of Hauser:
The Hironaka theorem on resolution of singularities http://www.ams.org/journals/bull/2003-40-03/S0273-0979-03-00982-0/home.html
The existence of resolution of ...

**7**

votes

**1**answer

416 views

### Proving a variety is not unirational

It is known that if a variety is unirational then it is rationally connected. However, there are no known examples of rationally connected varieties which are not unirational. In these notes, at the ...

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votes

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### Training towards research on birational geometry/minimal model program

Being a not yet enrolled independently supervised graduate student in mathematics, with prospects of applying to American graduate schools hopefully in a 1-2 years' time, I have a background of having ...

**7**

votes

**1**answer

266 views

### Is the number of minimal models finite

Let $X$ be a variety of general type.
Assume that $\dim X = 3$. In https://eudml.org/doc/164223 it is proven that $X$ has only finitely many minimal models (i.e., only $\mathbb Q$-factorial terminal ...

**7**

votes

**1**answer

424 views

### Does a birational involution of C^n always have a fixed point?

The title describes completely the question.
For n=1 it is an easy exercise. For n=2 the statement is still true, and depends on the classification of birational involutions of P^2 (see e.g. ...

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**0**answers

191 views

### Projectivity of flops

Say I have a projective (smooth, compact) irreducible symplectic variety $X$ over $\mathbb{C}$ and I perform a Mukai flop. It is well known that if the resulting variety $\widetilde{X}$ is Kahler, it ...

**6**

votes

**3**answers

452 views

### Why is the Hodge class of \bar{M_g} big and nef?

Let pi: \bar{Mg,1} \to \bar{M_g} be natural projection of compactified moduli stacks of curves and let omega be the relative dualizing sheaf. Then the Hodge class \lambda of \bar{M_g} is the first ...

**6**

votes

**1**answer

877 views

### Why is the inverse of a bijective rational map rational?

Let $f:X\to Y$ be a bijective rational map of an open dense subset $X$ of $\mathbb{C}\times\mathbb{C}$ onto an open dense subset $Y$ of $\mathbb{C}\times\mathbb{C}$. How to prove that the inverse map ...

**6**

votes

**2**answers

367 views

### Quotients of rational surfaces

Let $X$ be a projective surface defined over a field $k$ of characteristic $0$, and let $G$ be a finite group acting biregularly on $X$.
Assuming that $X$ is rational over $k$, is the quotient $X/G$ ...

**6**

votes

**1**answer

234 views

### Vanishing theorem for big divisors

Let $X$ be a projective, smooth variety over $\mathbb{C}$, and let $D$ be a irreducible, big Cartier divisor(notice, I do not assume nefness). Then is it true that $${\rm{H}}^1(X, K_X + D) = 0\quad?$$
...

**6**

votes

**1**answer

345 views

### Motivation behind the proof that $X^4+Y^4+Z^4+W^4$ is unirational

I'm trying to understand the proof that $V(x^4+y^4+w^4+z^4)$ in $\mathbb P^3_k$ is unirational for $k=\overline{\mathbb F_3}$.
The complete details are in the link so I just write a fast summary, ...

**5**

votes

**1**answer

415 views

### is Hasse principle a birational invariant?

...it is probably a very trivial question, but I am a beginner in arithmetics.

**5**

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**3**answers

591 views

### When does f-nef imply nef (after twisting?)

For a surjective morphism $f \colon X \to Z$ of smooth projective varieties, a line bundle $L$ on $X$ is $f$-nef if is nef on the fibers of $f$. More precisely, for every curve $C$ that is mapped to a ...

**5**

votes

**2**answers

333 views

### Hn(X, OX) = 0 for X birational to a regular affine variety?

It is a basic fact that $H^n(X, F) = 0$ if $X$ is noetherian affine, $n > 0$, and $F$ a quasi-coherent sheaf.
If $Y \to X$ is a blow-up of a smooth variety in a smooth center, then then ...

**5**

votes

**2**answers

596 views

### Is it possible to check two curves on birational equivalence by some computer algebra system?

I have two curves, for example hyperelliptic:
\begin{align}
&y^2 = x^6 + 14x^4 + 5x^3 + 14x^2 + 18, \\\\
&y^2 = x^6 + 14x^4 + 5x^3 + 14x^2 + 5x + 1
\end{align}
Is it possible to check them ...

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votes

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197 views

### $Aut(X)$ and $Bir(X)$ for Calabi-Yau 3-folds with $\rho(X)=1$

Let $X$ be a Calabi-Yau 3-fold with Picard number one. How can one show that the automorphism group $Aut(X)$ is finite and moreover coincides with the birational automorphism group $Bir(X)$?
It seems ...

**5**

votes

**1**answer

114 views

### Are there only finitely varieties of general type dominated by a given variety in the following sense

Let $X$ be a smooth projective complex algebraic variety. I believe the answer to the following question should be well-known. It is an instance of the Iitaka conjecture.
There are only finitely many ...

**5**

votes

**1**answer

779 views

### Minimal Model Program for surfaces over algebraically closed fields of characteristic p

Let $k$ be an algebraically closed field of characteristic $p>0$.
I have been trying to find out unsuccessfully if there is a mmp for algebraic surfaces over $k$. I know minimal surfaces are ...

**5**

votes

**2**answers

573 views

### Topology of the preimage of a point for degree one holomorphic maps

Let $M^n$ and $N^n$ be two compact complex (or complex projective) manifolds. Let $f: M\to N$
be a holomorphic map of degree one. How to prove that for each $x\in N$ the set
$f^{-1}(x)$ is ...

**5**

votes

**1**answer

455 views

### Can a birational morphism between smooth varieties be dominated by smooth blowup sequences?

Suppose $f:X\rightarrow Y$ is a birational morphism between smooth varieties and D is a snc divisor on $Y$. Can we find a smooth blowup sequence on $Y$ which dominates $f$ such that the preimage of D ...

**5**

votes

**1**answer

240 views

### A question on young persons guide to canonical singularities

In the Corollary at pag 407 of Young persons guide to canonical singularities there is a formula to compute the contributions $c_q(D)$ to Riemann-Roch of a divisor $D$ passing through a point $q\in ...